Tetraoctagonal tiling

In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.

Tetraoctagonal tiling
Tetraoctagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.8)2
Schläfli symbol r{8,4} or
rr{8,8}
rr(4,4,4)
t0,1,2,3(∞,4,∞,4)
Wythoff symbol 2 | 8 4
Coxeter diagram CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png or CDel node 1.pngCDel split1-84.pngCDel nodes.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node 1.png or CDel node.pngCDel split1-88.pngCDel nodes 11.png
CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node.png
CDel labelinfin.pngCDel branch 11.pngCDel 4a4b.pngCDel branch 11.pngCDel labelinfin.png
Symmetry group [8,4], (*842)
[8,8], (*882)
[(4,4,4)], (*444)
[(∞,4,∞,4)], (*4242)
Dual Order-8-4 quasiregular rhombic tiling
Properties Vertex-transitive edge-transitive

Constructions

There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1+], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1+,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1+,8,4,1+], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).

Four uniform constructions of 4.8.4.8
Name Tetra-octagonal tiling Rhombi-octaoctagonal tiling
Image Uniform tiling 84-t1 Uniform tiling 88-t02 Uniform tiling 444-t01 4242-uniform tiling-verf4848
Symmetry [8,4]
(*842)
CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node c3.png
[8,8] = [8,4,1+]
(*882)
CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node h0.png = CDel node c1.pngCDel split1-88.pngCDel nodeab c2.png
[(4,4,4)] = [1+,8,4]
(*444)
CDel node h0.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node c3.png = CDel label4.pngCDel branch c2.pngCDel split2-44.pngCDel node c3.png
[(∞,4,∞,4)] = [1+,8,4,1+]
(*4242)
CDel node h0.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node h0.png = CDel labelinfin.pngCDel branch c2.pngCDel 4a4b.pngCDel branch c2.pngCDel labelinfin.png or CDel nodeab c2.pngCDel 4a4b-cross.pngCDel nodeab c2.png
Schläfli r{8,4} rr{8,8}
=r{8,4}1/2
r(4,4,4)
=r{4,8}1/2
t0,1,2,3(∞,4,∞,4)
=r{8,4}1/4
Coxeter CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node h0.png = CDel node.pngCDel split1-88.pngCDel nodes 11.png CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png = CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node.png CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node h0.png = CDel labelinfin.pngCDel branch 11.pngCDel 4a4b.pngCDel branch 11.pngCDel labelinfin.png or CDel nodes 11.pngCDel 4a4b-cross.pngCDel nodes 11.png

Symmetry

The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.

Ord84 qreg rhombic til H2chess 248e

See also

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.

External links

Rhombitetraoctagonal tiling

In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,4}. It can be seen as constructed as a rectified tetraoctagonal tiling, r{8,4}, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling.

Snub tetraoctagonal tiling

In geometry, the snub tetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{8,4}.

Truncated tetraoctagonal tiling

In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.

*n42 symmetry mutations of quasiregular tilings: (4.n)2
Symmetry
*4n2
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
 
[ni,4]
Figures Uniform tiling 432-t1 Uniform tiling 44-t1 H2 tiling 245-2 H2 tiling 246-2 H2 tiling 247-2 H2 tiling 248-2 H2 tiling 24i-2
Config. (4.3)2 (4.4)2 (4.5)2 (4.6)2 (4.7)2 (4.8)2 (4.∞)2 (4.ni)2
Dimensional family of quasiregular polyhedra and tilings: (8.n)2
Symmetry
*8n2
[n,8]
Hyperbolic... Paracompact Noncompact
*832
[3,8]
*842
[4,8]
*852
[5,8]
*862
[6,8]
*872
[7,8]
*882
[8,8]...
*∞82
[∞,8]
 
[iπ/λ,8]
Coxeter CDel node.pngCDel 3.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel ultra.pngCDel node 1.pngCDel 8.pngCDel node.png
Quasiregular
figures
configuration
H2 tiling 238-2
3.8.3.8
H2 tiling 248-2
4.8.4.8
H2 tiling 258-2
8.5.8.5
H2 tiling 268-2
8.6.8.6
H2 tiling 278-2
8.7.8.7
H2 tiling 288-2
8.8.8.8
H2 tiling 25i-2
8.∞.8.∞
 
8.∞.8.∞
Uniform octagonal/square tilings
[8,4], (*842)
(with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries)
(And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-88.pngCDel nodes.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png
= CDel label4.pngCDel branch 11.pngCDel 4a4b-cross.pngCDel branch 11.pngCDel label4.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-88.pngCDel nodes 11.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node.pngCDel split1-88.pngCDel nodes 11.png
= CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel branch 11.pngCDel label4.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node 1.png
= CDel label4.pngCDel branch.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
H2 tiling 248-1 H2 tiling 248-3 H2 tiling 248-2 H2 tiling 248-6 H2 tiling 248-4 H2 tiling 248-5 H2 tiling 248-7
{8,4} t{8,4}
r{8,4} 2t{8,4}=t{4,8} 2r{8,4}={4,8} rr{8,4} tr{8,4}
Uniform duals
CDel node f1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node f1.png
H2chess 248b H2chess 248f H2chess 248a H2chess 248e H2chess 248c H2chess 248d H2checkers 248
V84 V4.16.16 V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16
Alternations
[1+,8,4]
(*444)
[8+,4]
(8*2)
[8,1+,4]
(*4222)
[8,4+]
(4*4)
[8,4,1+]
(*882)
[(8,4,2+)]
(2*42)
[8,4]+
(842)
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
= CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node.png
= CDel node h.pngCDel split1-88.pngCDel nodes hh.png
CDel node.pngCDel 8.pngCDel node h1.pngCDel 4.pngCDel node.png
= CDel label4.pngCDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 10.png
CDel node.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png
= CDel label4.pngCDel branch hh.pngCDel split2-44.pngCDel node h.png
CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node h1.png
= CDel node.pngCDel split1-88.pngCDel nodes 10lu.png
CDel node h.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node h.png
= CDel label4.pngCDel branch hh.pngCDel 2a2b-cross.pngCDel nodes hh.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png
Uniform tiling 444-t0 Uniform tiling 84-h01 Uniform tiling 443-t1 Uniform tiling 444-snub Uniform tiling 88-t0 Uniform tiling 54-t2 Uniform tiling 84-snub
h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4}
Alternation duals
CDel node fh.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node fh.png
Uniform tiling 88-t1 Uniform tiling 66-t1 Uniform dual tiling 433-t0 Uniform tiling 88-t2 Uniform tiling 54-t0
V(4.4)4 V3.(3.8)2 V(4.4.4)2 V(3.4)3 V88 V4.44 V3.3.4.3.8
Uniform octaoctagonal tilings
Symmetry: [8,8], (*882)
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png = CDel nodes 10ru.pngCDel split2-88.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png = CDel nodes 10ru.pngCDel split2-88.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png = CDel nodes.pngCDel split2-88.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png = CDel nodes 01rd.pngCDel split2-88.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node 1.png = CDel nodes 01rd.pngCDel split2-88.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node 1.png = CDel nodes 11.pngCDel split2-88.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png = CDel nodes 11.pngCDel split2-88.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node 1.png
H2 tiling 288-1 H2 tiling 288-3 H2 tiling 288-2 H2 tiling 288-6 H2 tiling 288-4 H2 tiling 288-5 H2 tiling 288-7
{8,8} t{8,8}
r{8,8} 2t{8,8}=t{8,8} 2r{8,8}={8,8} rr{8,8} tr{8,8}
Uniform duals
CDel node f1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node f1.png
H2chess 288b H2chess 288f H2chess 288a H2chess 288e H2chess 288c H2chess 288d H2checkers 288
V88 V8.16.16 V8.8.8.8 V8.16.16 V88 V4.8.4.8 V4.16.16
Alternations
[1+,8,8]
(*884)
[8+,8]
(8*4)
[8,1+,8]
(*4242)
[8,8+]
(8*4)
[8,8,1+]
(*884)
[(8,8,2+)]
(2*44)
[8,8]+
(882)
CDel node h1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png = CDel label4.pngCDel branch 10ru.pngCDel split2-88.pngCDel node.png CDel node h.pngCDel 8.pngCDel node h.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node h1.pngCDel 8.pngCDel node.png = CDel nodes 11.pngCDel 4a4b-cross.pngCDel nodes.png CDel node.pngCDel 8.pngCDel node h.pngCDel 8.pngCDel node h.png CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node h1.png = CDel node.pngCDel split1-88.pngCDel branch 01ld.png CDel node h.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node h.png = CDel nodes hh.pngCDel split2-88.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node h.pngCDel 8.pngCDel node.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 8.pngCDel node h.png = CDel nodes hh.pngCDel split2-88.pngCDel node h.png
= CDel node h0.pngCDel 4.pngCDel node h.pngCDel 8.pngCDel node h.png
Uniform tiling 88-h0 Uniform tiling 444-t0 Uniform tiling 88-h0 Uniform tiling 443-t1 Uniform tiling 88-snub
h{8,8} s{8,8} hr{8,8} s{8,8} h{8,8} hrr{8,8} sr{8,8}
Alternation duals
CDel node fh.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node fh.png CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node fh.png
Uniform tiling 88-t1 Uniform tiling 66-t1
V(4.8)8 V3.4.3.8.3.8 V(4.4)4 V3.4.3.8.3.8 V(4.8)8 V46 V3.3.8.3.8
Uniform (4,4,4) tilings
Symmetry: [(4,4,4)], (*444) [(4,4,4)]+
(444)
[(1+,4,4,4)]
(*4242)
[(4+,4,4)]
(4*22)
CDel label4.pngCDel branch 01rd.pngCDel split2-44.pngCDel node.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
CDel label4.pngCDel branch 01rd.pngCDel split2-44.pngCDel node 1.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node 1.png
CDel node h0.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node 1.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch hh.pngCDel split2-44.pngCDel node h.png
CDel node h0.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png
CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node h1.png
CDel node h0.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node h1.png
CDel label4.pngCDel branch hh.pngCDel split2-44.pngCDel node.png
CDel node h0.pngCDel 8.pngCDel node h1.pngCDel 4.pngCDel node.png
H2 tiling 444-1 H2 tiling 444-3 H2 tiling 444-2 H2 tiling 444-6 H2 tiling 444-4 H2 tiling 444-5 H2 tiling 444-7 Uniform tiling 444-snub H2 tiling 288-4 H2 tiling 344-2
t0(4,4,4)
h{8,4}
t0,1(4,4,4)
h2{8,4}
t1(4,4,4)
{4,8}1/2
t1,2(4,4,4)
h2{8,4}
t2(4,4,4)
h{8,4}
t0,2(4,4,4)
r{4,8}1/2
t0,1,2(4,4,4)
t{4,8}1/2
s(4,4,4)
s{4,8}1/2
h(4,4,4)
h{4,8}1/2
hr(4,4,4)
hr{4,8}1/2
Uniform duals
H2chess 444b H2chess 444f H2chess 444a H2chess 444e H2chess 444c H2chess 444d H2checkers 444 Uniform dual tiling 433-t0 H2 tiling 288-1 H2 tiling 266-2
V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V8.8.8 V3.4.3.4.3.4 V88 V(4,4)3

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